Lots of people get confused by this, but its true. A smaller sensor will have more depth of field given the same lens focal length, aperture, and distance to subject.
The web site tries to explain this, its based on the circle of confusion, which is different for small or large sensors. They use the term DSLR to mean a crop sensor.
If you frame the subjects the same, the depth of field will be the same, The depth of field gets larger as you backup with a crop camera in order to frame the subject the same.
With the point and shoot, if you frame the subject the same, the DOF will be the same.
Sorry, but only your very first sentence and your reason (circle of confusion, CoC) were correct - an accurate response to the OP's question (does a smaller sensor have shallower DoF when FL/aperture/distance are constant?). That's true, and the circle of confusion is the reason, and many people are confused by the whole concept.
DoF is determined by subject magnification and aperture (the physical diameter of the iris diaphragm), and the CoC enables practical comparisons of DoF. Focal length and subject distance affect subject magnification, therefore they affect DoF.
If you match subject framing, the smaller the sensor, the deeper the DoF. That's generally the most useful type of comparison, because we are usually interested in comparing hope the same picture."
The crop sensor is just looking at a smaller part of the image circle that the lens is projecting so it should be the same as cropping a ff image. Why would the dof change if the fl and distance are constant?
Cropping an image does
change the DoF (pauses until the sound of heads exploding fades
). Well, sort of.
PBD summed it up nicely - fundamentally, DoF is very subjective. Since DoF is really about differential sharpness in an image, it's affected by more than just the parameters of how the image is captured. Cropping and enlargement, viewing distance (for the monitor or print), and even the viewer's visual acuity all affect DoF. CoC attempts to control for those post-capture factors by assuming a specific, fixed print size, viewing distance, and viewer acuity.
The fixed print size means that an image from a smaller sensor must be enlarged more, i.e., the original magnification was relatively lower. Similarly, cropping an image in post (where aperture, focal length, and distance cannot be changed) results in a shallower DoF given the assumptions of CoC (i.e., the cropped image would be enlarged to the same final output size).